A FriidIy to LA cti L Si , h 41' d1 I OND It - iv tot IE 10 I b n i W I T L D A J ,, - _ K' Tai ngay!!! Ban co the xoa dong chu nay!!! 16990026332821000000 A Friendly Introduction to Analysis Single and Multivariable A Friendly Introduction to Analysis Single and Multivariable Second Edition Witold A J Kosmala Appalachian State University PEARSON Prentice hail Upper Saddle River, New Jersey 07458 Library of Congress Cataloging-in-Publications Data Kosmala, Witold A.J A friendly introduction to analysis; single and multivariable 2nd ed.IWitold A.J Kosmala p cm Includes bibliographical references and index, ISBN 0-13-045796-5 Calculus CIP data available Executive Acquisitions Editor: George Lobe!! Executive Editor-in-Chief: Sally Yagan Vice PresidentlDirector of Production and Manufacturing: David W Riccardi Production Editor: Debbie Ryan Senior Managing Editor: Linda Mihatov Behrens Assistant Managing Editor: Bayani Mendoza de Leon Executive Managing Editor: Kathleen Schiaparelli Assistant Manufacturing Manager/Buyer: Michael Bell Manufacturing Manager: Trudy Pisciotti Marketing Manager: Halee Dinsey Marketing Assistant: Rachel Beckman Cover Designer: Bruce Kenselaar Creative Director: Jayne Conte Director of Creative Services: Paul Belfanti Art Editor: Thomas Benfatti Editorial Assistant: Jennifer Brady Cover Image: ©Witold A J Kosmala PEAR SON © 2004, 1999 Pearson Education, Inc Pearson Prentice Hall Pearson Education, Inc Upper Saddle River, NJ 07458 All rights reserved No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher Pearson Prentice Hall© is a trademark of Pearson Education, Inc Printed in the United States of America 1098765 ISBN 0-13-045796-5 Pearson Education, Ltd., London Pearson Education Australia PTY Limited, Sydney Pearson Education Singapore, Pte., Ltd Pearson Education North Asia Ltd, Hong Kong Pearson Education Canada, Ltd., Toronto Pearson Education de Mexico, S.A de C.V Pearson Education - Japan, Tokyo Pearson Education Malaysia, Pte Ltd Pearson Education, Upper Saddle River, New Jersey To my wife, Vanessa, and my children, Konrad, Doria, Alma, and Henryk Contents Preface xi Introduction .1 1.1 ** Algebra of Sets 1.2* Relations and Functions 1.3 * Mathematical Induction 20 1.4* Proof Techniques 31 1.5 * Inverse Functions 36 1.6* Finite and Infinite Sets 40 1.7* Ordered Field and a Real Number System I.8* Some Properties of a Real Number 48 1.9* Review 55 1.10* Projects 57 Part Fibonacci Numbers 57 Part Lucas Numbers 61 Part Mean of Real Numbers 61 43 Sequences 2.1 2.2 2.3 2.4 2.5 2.6 Convergence 65 Limit Theorems 74 Infinite Limits 81 Monotone Sequences 88 Cauchy Sequences 97 Subsequences 106 2.7* Review 111 2.8* Projects 113 Part The Transcendental Number e Part Summable Sequences 115 *Optional sections are marked with an asterisk vii 113 65 Contents viii Limits of Functions 3.1 3.2 3.3 Limit at Infinity 117 Limit at a Real Number Sided Limits 133 3.4* Review 3.5 * Projects Part Part 4.2* 4.3 4.4 45* 4.6* 142 144 Monotone Functions Continued Fractions 144 145 Continuity of a Function 148 Discontinuity of a Function 156 Properties of Continuous Functions 162 Uniform Continuity 168 Review 176 Projects 179 Part Compact Sets 179 Part Multiplicative, Subadditive, and Additive Functions 5.2 5.3 5.4 55* 5.6* 5.7* 6.2 6.3 6.4 6.5 6.6* 6.7* 6.8* 183 Derivative of a Function 184 Properties of Differentiable Functions 193 Mean Value Theorems 201 Higher Order Derivatives 209 L' Hopital's Rules 220 Review 228 Projects 232 Part Approximation of Derivatives 232 Part Lipschitz Condition 234 Part Functions of Bounded Variation 235 Part Absolutely Continuous Functions 237 Part Convex Functions 238 Integration 6.1 gas 180 Differentiation 5.1 125 Continuity 4.1 117 Riemann Integral 241 Integrable Functions 246 Properties of the Riemann Integral 250 Integration in Relation to Differentiation Improper Integral 265 Special Functions 274 Review 283 Projects 286 256 Zao Contents ix Part Part Part Part Infinite Series 7.1 7.2 7.3 7.4 7.5* 7.6* Convergence 294 Tests for Convergence 303 Ratio and Root Tests 311 Absolute and Conditional Convergence Review 325 Projects 327 Part Summation by Parts 327 Part Multiplication of Series 329 Part Infinite Products 330 Part Cantor Set 332 8.2 8.3 8.4 8.5 8.6 8.7* 8.8* 9.2* 9.3* 9.4 9.5* 9.6 9.7 9.8* 9.9* 334 Pointwise Convergence 335 Uniform Convergence 341 Properties of Uniform Convergence 347 Pointwise and Uniform Convergence of Series 351 Power Series 360 Taylor Series 370 Review 375 Projects 378 Part Limit Superior 378 Part Irrationality of a 380 Part An Everywhere Continuous but Nowhere Differentiable Function 381 Part Equicontinuity 381 Vector Calculus 9.1 * 294 317 Sequences and Series of Functions 8.1 Wallis's Formula 286 Euler's Summation Formula 287 Laplace Transforms 289 Inverse Laplace Transforms 292 Cartesian Coordinates in R3 Vectors in J13 383 385 Dot Product and Cross Product Parametric Equations 395 Lines and Planes in & 442 Vector-Valued Functions 405 Arc Length 416 Review 423 Projects 425 Part Inner Product 425 388 383 Greek Alphabet A a B fi F y A o E Z H E, E @ 8, z9 I K A M t j x ) i alpha beta gamma delta epsilon zeta eta theta iota kappa lambda mu N v flu O H P o it E a T r omicron pi rho sigma tau upsilon phi chi psi omega xi Y X 'P SZ 560 p v , rp X yi w Index of Symbols { }, E, 1(x),14 Li, n f , 14 n Ak, k=1 N, Z, Q,2 Q,2 [x], Lxi, 15 n 15 U f1k, k=1 aR, 15 00 U Ak, E, 16, 21 k;1 00 fng,17 U Ak, k=1 [a, b], (a, b), 3, f n, 17 Ax B,6 fog, 17 J126 , n [a, b), d(A, B), (a, b], P(A), (a, x), graph(f ), ,21 k=1 (a, ox), (-x, a], (-oo, a), oo, +oo, oo, (-oo, oo), f (a), (), 23 C, 23 D1, n!,23 R1, p f (A), 10 ^- p, 32 f :A - B, 10 , 32 (C), 10 *, 32 (0, c), f =, lxi, 13 if, P (k), 22 ,33 sup f , 14 b, 33 sup f (x), 14 2,33 xEA f, 36 ,3 C, C, inf f, 14 0, n, q,32 A f (x), 14 B , 40 Fn, 57 xEA U, -, 14, 32 L0, 61 \,3 maxf,14 H(a, b), 62 561 Index of Symbols 562 G (a, b), 62 A (a, b), 62 Q (a , b), 62 Ja (f ), 159 Aa, 179 aEG n n {an }, 65 fl, 181 {a0}°_p, 66 i=1 n - o0 ff(n), 181 ,u(n), 181 fi(n), 181 lirn an = 00 an = ff,244 fab f , 245 acG E0, 62 f 'f,244 y , 255, 278 f' (a), 184 f f (x)dx, 259 f00° 1,266 b ff,266 f 0(b0), 85 f', 186 n!!, 97 Ox, 186 af(n), 106 oy, 186 ank , 107 d, lim sup an, 107 d , 186 C(x), 273 y', 186 Si(x), 273 jy, 186 Ci(x), 273 urn an , 107 fl- 00 lim 1nf an, 108 fl-4 00 lirn an , 108 f " (a), 209 210 n -* 00 d-dx e, 113,380 f (4) (x ), 210 (C, 1), 115 I E C1, 211 (C, 2) , 116 f (n) (x), 211 lim f(x)=L, 117 lim f (x) = +00, 121 X- -oo f(x)=L, 121 lim f (x), 121 g, 00°-0 , oo, 00 - 00,100, 00°, 0°, 223 268 T (x ), 274 ji(x), 277 B(x), 278 erf(x), 278 erfc(x), 278 J(x), 278 f (x), 232 Jp (x), 279 f (x ) , 232 Yp (x), 279 X-4 ±00 0(h), 232 Iirn f (x) = L, 126 A0 f (x), 232 x m f (x) = +00, 130 o0 f (x), 232 x -a CPV, S(x), 273 186 , f , 268 Pn (x), 279 Hn (x ), 279 of (x), 233 279 x - a+ 0(h2), 233 280 f (a+) , 133 lirn f (x), 133 on f , 233 1im 1(x) = L, 133 f (a-) , 133 V f (a, b), 236 J9 (x ), 282 f+ (a), 239 Sf{ f (t)}, 289 f' (a), 239 f (t) * g (t ), 291 x-4a+ IPI, 241 5f { F (x) }, 292 ex, exp(x), 135 Mk (x°, x 1, ) , 145 U (P, f), 241 lim f (x) = +oo, 133 241 sins x, 153 L(P, f), 241 S(P, f), 241 sgn x, 154 LIxk, 241 (xo, x l, , x0 ), 145 a0 a,294 294 k=p ak, 294 c(x), 311,331 Index 563 aR +, 320 an , 320 S, 320 Sn -, 320 Sn *, 320 5ij, 329 R ak, 330 k=1 = ak, 330 k!1 = fb r(t)dt, 411 f a r(t)dt, 411 Du f , 463 [u, v, w], 415 481 481 L(y), 416 LQ, 416, 417 (u, v), 425 [r, 9], 426 B(a; r), 433 int S, 434 a ,463 M11, 481 LfR f,481 ffR f,482 fIR f,482 fb fed f (x, y)dy dx, 484 flak, 330 S, 434 ff1,493 {fn(x)}, 335 S',434 fc f ds, 496 ao 351 f( x),351 k=1 i, 375 (a, b, c), 84 R3,384 PQ, 385 v, v, 385 i, j, k, 385 0, 0, 385 (a, b, c), 385 II u II , 386 u v, 388 proj3 u, 391 u x v, 392 (f (t), g(t), h(t)), 396 a S, 434 lim (x, y)-*(a,b) f (x, y), 439 f&' b), 448 fy(a, b), 448 ax ,4.48 Ix, f y, 449 a , ,449 y af 1i:0o ayn , , fxy f yx fyy 450 D,452 r(t), 397 Dx, Dy, 452 Eh, 453 lim F(t), 406 Oh, 453 y (t), 396 t-+ro Af(t), 410 Dn, 453 V2 , 454 V2u, Vu, 454 v4, 455 V f, 458 AN(t), 410 V,459 '(t) , 407 T (t), 408 V (t), 408 A (t), 408 F(x, y), 500 fcP1F,501 fC Mdx + Ndy, 502 f7=a F ds, 502 c F - Nds, 503 fc Mdy - Ndx, 503 curl F, 515, 518 div F, 519 J(u, v), 526 Index A Abel, Niels, 269 Abel's identity, 328 Abel's limit theorem, 379 Abel's test, 269, 329, 354 Abel's theorem, 330, 363 absolute extremum, 196 maximum, 14, 196 minimum, 14, 196 value, 13 absolutely continuous, 237 convergent, 319, 353 improper integrable, 268 uniformly convergent, 353 abundant number, 181 acceleration vector, 408 accumulation point, 98, 434 additive function, 182 identity, 43 inverse, 43 Agnesi, Maria, 401 algebraic, 15, 46 alternating harmonic series, 319 alternating sequence, 317 alternating series, 318 alternating series test, 318 analytic, 372 angle between vectors, 387 annihilate, 455 annular region, 517 antiderivative, 205, 259, 411 arc length, 416 arc length parameter, 420 Archimedean order property, 45, 92 Archimedes, 45 arcwise connected, 436 arithmetic average sequence, 115 Arzela, Cesare, 348 Arzela's theorem, 348 Arzela-Ascoli theorem, 382 Ascoli, Giulio, 382 associative, 4, 43 astroid, 199,401 asymptote even vertical, 134 horizontal, 118 oblique, 122 odd vertical, 134 vertical, 134 average, 62 average value, 253 axiom, 20, 43 completeness, 45, 97 Dedekind's, 45 axis, 3, B backward difference, 232 basic open set, 434 Bell, Eric, 277 Bellavitis, Giusto, 201 Bernoulli, Jacob, 373, 478 Bernoulli, Johann, 30, 399, 401 Bernoulli equation, 478 Bernoulli's inequality, 30 Bernoulli's numbers, 373 Bessel, Friedrich, 279 Bessel's equation, 282 Bessel's function, 279 beta function, 278 Bianchi, Luigi, 485 big oh, 85 biharmonic function, 455 bijection, 11 Binet, Jacques, 59 Binet form, 59 binomial coefficient, 23 expression, 25 theorem, 24 Bolzano, Bernhard, 97 Bolzano's intermediate value thm, 163 Bolzano-Weierstrass theorem, 99, 108, 435, 438 Bombelli, Rafael, 60 Bonnet, Pierre, 253 Bonnet's mean value theorem, 253 564 Index Bore!, Emil, 180 bound, least upper, 13, 45 bound, upper, 13 boundary, 435 boundary point, 435 bounded, 13, 14, 70, 434, 438, 504 above, 13, 14 away from zero, 71, 167 below, 13, 14 pointwise, 339 uniformly, 339, 351 bounded variation, 236, 325 Bowditch, Nathaniel, 414 Bowditch curve, 414 box product, 415 brachistochrone, 399 Brouwer, Luitzen, 165 Brouwer's fixed point theorem, 165 Buniakovski, Victor, 53 C Cantor, Georg, 41 Cantor function, 431 Cantor intersection theorem, 435 Cantor set, 332 cardinality, 40 cardioid, 428 Cartesian coordinate system, Cartesian product, 6, 384 Cassini, Giovanni, 199 catenary, 373 Cauchy, Augustin, 53 Cauchy criterion, 299, 330, 346, 353 Cauchy density function, 271, 340 Cauchy principal value, 268, 270 Cauchy product, 329 Cauchy sequence, 100,413 Cauchy's condensation test, 308 Cauchy's function, 217, 227 Cauchy's mean value theorem, 204 Cauchy's ratio test, 312, 315 Cauchy's root test, 314 Cauchy-Hadamard theorem, 363, 379 Cauchy-Riemann equations, 471 Cauchy-Schwarz inequality, 53, 255, 311, 390,425 Celsius, Anders, 39 center, 384, 433 Cesaro, Ernesto, 115 Cesaro convergent, 115 chain rule, 194, 465, 466 change, 186 change of variables theorem, 261, 526 characteristic equation, 27, 476 characteristic function, 340 Chebyshev, Pafnuti, 280 565 Chebyshev equation, 283 Chebyshev polynomials, 280 circle, 8, 428 circular helix, 422 circulation, 503, 519 circulation density, 506 circumference, 421 Clairaut, Alexis, 451 Clairaut's theorem, 451 closed, 162, 399, 434, 503 closed form, 257 closed region, 435 closure, 43, 434 cluster point, 98 coefficient, 14 commutative, 4, 43 compact, 179 comparison test, 266, 305 comparison theorem, 82 complement, 3, 435 complementary error function, 278 complete ordered field, 43 completely multiplicative, 180 completeness axiom, 45, 97 completing the square, 54 complex conjugates, 476 complex number, 375 component, 386 component function, 405, 500 composite function, 17 composition, 17 concave, 238 down, 210 strictly, 230 up, 210 conditionally convergent, 319 conditionally improper integrable, 269 conjecture, 21, 31 connected, 435 connected by a path, 436 conservative vector field, 505, 519 constant, 12, 15, 69 content zero, 492 continued fraction, 60, 145 finite, 145 infinite, 145 simple, 145 continued power, 156 continued root, 227, 317 continuous, 148, 397, 407, 442 continuous extension, 157 continuous from the right, 150 continuously differentiable, 211, 408, 460 contour line, 439 contraction, 173 contraction principle, 102 Index 566 contractive, 102, 173, 445 contradiction, 32 convergent, 335 converges, 66, 266, 267, 295, 330, 335, 351, 438 converges absolutely, 268, 319 converges conditionally, 269, 319 converges in the mean, 339 converges pointwise, 335, 351 converges uniformly, 341 convex, 238, 438 logarithmically, 239 midpoint, 28, 238 strictly, 238 convolution, 291 convolution theorem, 291 coordinate, coordinate planes, 384 Copernicus, Nicolaus, 40 corollary, 31 cosine integral, 273 Cotes, Roger, 114 countable, 40 countably infinite, 40 counterclockwise circulation, 513 counterexample, 31 covers, 492 critical point, 197, 464 cross product, 392 cross section, 450 cumulative distribution function, 278 curl, 506, 511, 518 curl integral, 513 curl test, 516 curve, 8, 439 curvilinear coordinate system, 426 cusp, 398 cycloid, 398 cylinder, 404 cylindrical coordinates, 431 D d' Alembert, Jean, 312 d' AIembert's complete solution, 472 Darboux, Jean, 209 Darboux's intermediate value theorem, 209 De Moivre, Abraham, 59 De Moivre's formula, 375 De Morgan, Augustus, De Morgan's law, decreasing, 12, 89 eventually, 89 eventually strictly, 89 strictly, 89 Dedekind, Julius, 45 Dedekind's axiom, 45 Dedekind's test, 359 deficient number, 181 defined, 10 definite integral, 245, 265 degenerate circle, directed line segment, 385 sphere, 384 degree, 14 dense, 46 denumerable, 40 derivative, 183, 184, 186, 458 derived set, 434 Descartes, Rene, Descartes' rule of signs, 50 difference, 386 central, 233 first backward, 232 first forward, 232 quotient, 184 differentiable, 184, 186, 407, 408,458 differential operator, 452, 453 differentiation, 187 digamma function, 277 Dini, Ulisse, 349 Dini's theorem, 349, 352, 469 directed line segment, 385 directed set, 90 direction, 387, 398, 420 directional derivative, 463 Dirichlet, Peter, 129 Dirichlet's formula, 495 function, 129, 191, 249 test, 269, 328, 354 disconnected, 435 disconnection, 438 discontinuity, infinite, 158 jump, 159 missing point, 157 nonremovable, 159 of the first kind, 159 of the second kind, 159 oscillating, 160 removable, 158 simple, 159 discontinuous, 152 discriminant, 48 disjoint, 4, 436 distance, 51, 434 distance formula, 7, 384 distinct, distributive, 4, 43 divergence, 513, 519 divergence integral, 513 divergence test, 296 Index diverges, 68, 266, 267, 295, 330, 338 to +oo, 81 to 0, 330 divisible, 23 division algorithm, 145 divisor, 23 divisor function, 181 does not exist, 10, 81 domain, 10, 436, 439 first type, 494 natural, 10 restricted, 10 second type, 494 dominant term, 135 dot product, 388 double factorial, 97 double root, 213 Du Bois-Reymond, Paul, 359 Du Bois-Reymond's test, 359 Duhamel, Jean, 417 Duhamel's principle, 417 dummy variable, 65, 117, 244, 295 E e, 113,429 eccentricity, 429 Einstein, Albert, 32 element, ellipsoid, 404 elliptic cone, 404 elliptic equation, 454 elliptic paraboloid, 404 empty set, epicycloid, 428 equal, equals, equicontinuous, 382 equipotent, 44 equivalent, 40, 385, 419 error function, 278 essentially equivalent, 529 Euclid, Euclidean algorithm, 145 Euclidean distance formula, 7, 384 Euclidean inner product, 388 Euler, Leonhard, 114 Euler's ¢5-function, 181 constant, 255, 278 formula, 375 number, 113 numbers, 373 product for zeta function, 331 summation formula, 287 theorem, 471 even, 2, 12 567 even vertical asymptote, 134 eventually, 72 exists, 10 explicit, 25 exponential function, 135 exponential order, 289 extension, 157 extreme value theorem 163, 180 extremum, 14 F factor, 23 factored, 48 factorial, 23 double, 97 generalized, 275 factorial sequence, 94 Fahrenheit, Gabriel, 39 family of sets, 179 Fermat, Pierre de, 21 Fermat's last theorem, 21 Fibonacci, Leonardo, 57 Fibonacci numbers, 57 Fibonacci sequence, 57 field, 43 complete ordered, 43,45 inverse square, 510, 521 ordered, 44 scalar, 500 finite, 40, 145 first derivative test, 206 first octant, 384 first order linear equation, 477,478 fixed point, 106, 165 fixed point theorem, 165 floor bracket, 15 flow, 500 flow integral, 503 flow line, 500 flux, 503, 519 folium of Descartes, 200 forward and backward induction, 27 forward difference, 232 Fourier, Jean, 289 Frechet, Maurice, 433 Fresnel, Augustin, 273 Fresnel cosine integral, 273 Fresnel sine integral, 273 Fubini, Guido, 485 Fubini's theorem, 485 function, absolute value, 13 algebraic, 15 arithmetical, 15 bijective, 11 bounded variation, 236 Index 568 Cauchy density, 271, 340 composite, 17 constant, 12, 15 cubic, 15 decreasing, 12 even, 12 exponential, 135 gamma, 274 greatest integer, 15 increasing, 12 injective, 11 integer ceiling, 15 integer floor, 15 linear, 15 multiplicative, 180 number-theoretic, 15 postage, 16 quadratic, 15 rational, 15 real-valued, 10 single-valued, 10 surjective, 11 tax, 16 transcendental, 15, 135 trigonometric, 12 well-defined, 10 Witch of Agnesi, 340, 401 zero, 14 functional values, 10 fundamental period, 174, 399, 414 ratio test, 311 theorem of calculus, 257, 412, 505 G Gabriel's horn, 270 Galileo Galilei, 40 gamma function, 274 Gauss, Karl 147 Gauss's divergence theorem, 519 general term, 65 generalized factorial function, 275 generalized mean value theorem, 204 generalized Pythagorean theorem, 426 generalized Rolle's theorem, 220 geometric mean, 62 mean sequence, 156 progression, 298 sequence, 94 series, 297 sum, 23, 74 Gerschgorin, Sergi, 50 Gerschgorin's circle theorem, 50 global maximum, 14, 196 golden number, 60 gradient, 458, 470 gradient vector field, 505 Grandi, Luigi, 428 graph, greatest integer function, 15 greatest lower bound, 14 Green, George, 511 Green's identity, 521 Green's theorem, 511, 513 Gregory, James, 335 Gregory's series, 338 grid, 481 H Hadamard, Jacques, 363 Hamilton, William, 386 Hankel, Hermann, 289 harmonic, 62, 72, 297, 472 harmonic function, 454, 520 Hausdorff, Felix, 433 Heine, Edward, 180 Heine-Borel theorem, 180 helix, 422 Hermite, Charles, 47 Hermite polynomials, 429 Hermite equation, 283 higher order, 84 Hilbert, David, 46 Hilbert number, 46 Hire, Philippe de la, 431 Holder, Ludwig, 207 Holder's inequality, 207, 256 homogeneous, 209, 471, 475, 479 horizontal asymptote, 118 horizontal line test, 11 horizontal point of inflection, 211 Huygens, Christiaan, 373, 399 hyperbolic paraboloid, 404 hyperboloid, 404 hyperharmonic series, 304 hypocycloid, 401 hypothesis, 31 I idempotent, identically zero, 14 identity function, 14 if and only if, image, 8, 10 imaginary number, 375 implicit differentiation, 198 implicit function theorem, 469 implicitly, 25, 469 improper integral, 265, 266 increasing, 12, 89 eventually, 89 Index eventually strictly, 89 strictly, 89 increment, 186 indefinite integral, 259 indefinite Riemann integral theorem, 258 independence of parametrization, 504 indeterminate forms, 138, 220 index, 65 indexing set, 179, 434 induction hypothesis, 22 induction, 22 inductively, 25 inequality, arithmetic-geometric mean, 219 Bernoulli's, 30, 207 Cauchy-Schwarz, 53, 255, 311, 390, 425 Holder's, 207, 256 Jensen's, 28, 219, 285 Minkowski's, 54, 256, 311, 425 triangle, 51, 390, 425 infimum, 14 infinite, 40, 145 countably, 40 discontinuity, 158 limit, 81, 121 product, 330 series of functions, 351 series, 294 infinitely many times continuously differentiable, 211 infinity, initial conditions, 27 initial point, 385 injection, 11 inner product, 425 integer, integrable, 244, 411, 481, 493 integral curve, 500 integral sign, 245 integral test, 303 integrand, 245 integrating, 245 integrating factor, 476, 529 integration, 245 integration by parts, 258 interior, 434, 516 interior point, 434 intermediate value property, 163 intermediate value theorem, 163, 164, 209, 447 Bolzano's, 163 Darboux's, 209 Weierstrass, 164 intersection, interval, interval of convergence, 362 569 inverse, 10, 36 additive, 43 function, 17 function theorem, 197 image, 10 Laplace transform, 292 multiplicative, 43 operator, 477 square field, 510, 521 inverses of each other, 38 inverted cycloid, 398 invertible, 36 irrational number, irrotational, 503 isochrone, 373 isolated point, 149, 442 iterated integral, 484 iterated limits, 441 J Jacobi, Karl, 526 Jacobian, 526 Jensen, Lohan, 28 Jensen's inequality, 28, 219, 285 Jordan, Marie, 492 Jordan content zero, 492 Jordan measurable, 492 K Khayyam, Gmar, 24 Kirchhoff, Gustav, 519 Kronecker, Leopold, 329 Kronecker's delta, 329 Kummer, Ernst, 329 L L'Hopital, Guillaume, 220 L'Hopital's rule, 220, 222 Lagrange, Joseph, 186 Lagrange's first identity, 395 mean value theorem, 202 second identity, 395 Laguerre, Edmond, 279 Laguerre polynomials, 279 Lambert, Johann, 46 Laplace, Pierre, 289 Laplace transform, 289 Laplace's equation, 454, 520 Laplacian, 454, 459, 516, 520 in cylindrical coordinates, 522 operator, 454 in polar coordinates, 471 in spherical coordinates, 522 law of cosines, 389 leading coefficient, 15 Index 570 least upper bound, 13, 45 Lebesgue, Henri, 182 left-hand limit, 133 Legendre, Adrien, 279 Legendre equation, 286 Legendre polynomials, 279 Leibniz, Gottfried, 183 Leibniz rule, 217, 488, 490 Leibniz's alternating series test, 318 lemma, 31 lemniscate, 199, 428 length, 416 Lerch, Mathias, 293 Lerch's theorem, 293 level curve, 439 limacon, 428 of Pascal, 428 limit, 66, 117, 121, 126, 335, 351, 406, 439 comparison test, 266, 272, 306 exists, 81 inferior, 108 point, 107 from the right, 133 superior, 107 limiting function, 335 limiting value, 66 117, 441 Lindemann, Ferdinand, 47 line, 15, 402, 428 line integral, 496, 497 of scalar field, 501 of vector field, 501 w.r.t , 497 w.r.t arc length, 498 line segment, 15, 385 linear combination 176, 387, 475 differential equation, 477, 478, 531 function, 15 operator, 453 property, 250, 266, 289, 299, 504 Liouville, Joseph, 47 Lipschitz, Rudolf, 172 Lipschitz condition, 173, 234, 445 Lipschitz constant, 172, 234, 445 Lipschitz function, 172, 234, 445 Lissajous, Jules, 414 Lissajous figure, 414 local extremum, 195 maximum, 195 minimum, 195 logarithmic -series, 310 logarithmic differentiation, 199 logarithmic function, 135 logarithmically convex, 239 lower bound, 13 Darboux sum, 241 double integral, 481 integral, 244 limit, 108 sum, 241, 481 Lucas, Edouardo, 61 Lucas numbers, 61 M Maclaurin, Colin, 322 Maclaurin series, 370 magnification factor, 526 magnitude, 385, 386 mapping, 8, 525 maps, 10 Mascheroni, Lorenzo, 255 Mascheroni's constant, 255 mathematical induction, 22 maximum, 14, 196 mean value theorem, 202, 204, 467 Bonnet's, 253 Cauchy's, 204 first, 253 for double integrals, 484 for integrals, 253 generalized, 204, 265 Lagrange's, 202 second, 253, 265 mean, 61 arithmetic, 62 arithmetic-geometric, 96 arithmetic-harmonic, 97 exponential of order, 62 geometric, 62 harmonic, 62 quadratic, 62 weighted, 64 mean-square convergence, 339 Me11in, Robert, 289 member, Mertens, Franz, 329 Mertens's theorem, 329 mesh of partition, 241 midpoint convex, 28 minimum, 14, 196 Minkowski, Hermann, 54 Minkowski's inequality, 54, 256, 311,425 minus, minus infinity, missing point discontinuity, 172 mixed partials, 450 Mobius, August, 181 Mobius function, 181 monotone, 12, 89 eventually, 89 eventually strictly, 89 Index strictly, 89 multiple, 23 multiplicative, 180 identity, 43 inverse, 43 N n times continuously differentiable, 211 n times differentiable, 211 nth derivative, 210 nth term test, 296 natural logarithm, 114 natural number, necessary, 32 negative number, neighborhood, 98, 434, 439 deleted, 100439, 440 punctured, 98, 439 nested, 435 Neumann function, 279 Neumann, Karl, 279 Newton, Isaac, 95 Newton's binomial theorem, 24 Newton-Raphson method, 95 Nobel, Alfred, 32 Nobel Prize, 32 nondecreasing, 89 nonincreasing, 89 nonnegative number, nonpositive number, norm, 241, 425, 481 norm function, 447 noun of partition, 241 normal acceleration, 411 normal line, 199, 404, 470 normalized, 387 normally convergent, 353 null set, number of elements, 40 oblique asymptote, 122 octant, 384 odd, 2, 12 number, vertical asymptote, 134 one-dimensional, one-sided, 133 one-to-one, 11, 525 onto, 10 open, 162, 434 ball, 433, 439 cover, 179 disk, 434 region, 436 set, 162 571 sphere, 434 operationally equivalent, 453 operator, 452 central difference, 453 commutative, 453 differential, 453 forward, 453 inverse, 477 inverse Laplace, 453 Laplace, 453 Laplacian, 454,459 linear, 453 shift, 453 unit, 453 vector differential, 459 opposite direction, 387 order, 14 exponential, 289 higher, 84 same, 84 ordered field, 44 pair, triple, 84 orientation, 398,420 origin, orthogonal, 387, 425 orthogonal projection, 391 oscillatory, 52, 81 outward flux, 513 overbar, 244 P p-periodic, 174, 399, 414 p-series, 304 parabola, 15 parallel, 16, 387 parameter, 274, 396 parametric equations, 396 parametrization, 395, 396 parity, 35 partial antidifferentiation, 453 partial derivative, 448 partial differentiation, 449 partial integration, 454 partition, 235, 241,480 Pascal, Blaise, 25 Pascal, Etienne, 428 Pascal's triangle, 25, 26 path, 436 path independent, 506, 507 pathwise connected, 436 perfect number, 181 period, 174, 399, 414 periodic, 174, 399, 414 perpendicular, 16, 387 petal curve, 428 Index 572 piecewise continuous, 160 piecewise smooth, 397, 498 plane, Plato, plus infinity, Poincare, Henri, 47 point, point of accumulation, 98 point of inflection, 211 pointwise bounded, 339 pointwise limit function, 335 pointwise sum, 351 polar axis, 426 polar coordinates, 426 polar curve, 427 pole, 426 polynomial, 14 positive number, positively oriented, 511 postulate, 20 potential function, 506, 519, 528 potential theory, 454 power rule, 199 power series expansion, 360 power series representation, 360 power series, 360 power set, preimage, 10 prime, principle of mathematical induction, 21 product rule, 193 projection, 391 projection function, 446,447 proper subset, psi function, 277 punctured neighborhood, 98 punctured plane, 514 pure partial derivatives, 450 Pythagoras of Samos, Pythagorean theorem, Q quadratic formula, 48 function, 15 mean, 62 quadric surfaces, 404 quotient rule, 193 R racetrack principle, 206 radian measure, radius of convergence, 362 radius, 433 range, 8, 439 Raphson, Joseph, 95 rate of convergence, 85 rate, same, 84 ratio comparison test, 311 ratio test, 84, 312, 379 C auchy's, 312, 315 d' Alembert's, 312 fundamental, 311 generalized, 312 rational function, 15 rational number, rational root theorem, 49 real number, rearrangement, 321 reciprocal, 17 rectangular coordinate system, rectifiable, 418 recurrence, 25 recursion formula, 25 recursively, 25 refinement, 241, 481 reflexive, 40 region, 435 regrouping, 320 relation, relative extremum, 195 maximum, 195 minimum, 195 relatively prime, 33 remainder, Cauchy's form, 214,264 Lagrange's form, 214 term, 214 represent, 351, 360 rhodonea curve, 428 rhombus, 394 Riemann, Georg, 240 Riemann integrable, 244, 411, 481, 493 Riemann integral, 244, 482 Riemann sum, 241, 430, 481, 497 right continuous, 150 right-hand limit, 133 right-hand rule, 392 right-handed system, 384 Roberval, Gilles de, 428 Rolle, Michel, 201 Rolle's theorem, 201 generalized, 220 root test, 313, 379 Cauchy's, 314 root, 14, 212 double, 213 of multiplicity m, 213 simple, 213 S saddle, 404 Index same direction, 87 same order, 84 same rate, 84 sandwich theorem, 78, 120 sawtooth function, 161 scalar, box product, 415 curl, 506 field, 500 function, 407 multiplication, 386 triple product, 415 Schlomlich, 4skar, 371 Schwarz, Hermann, 53 secant line, 184 second derivative, 209 derivative test, 220 fundamental theorem of calculus, 259 order linear equation, 475 partial, 451 partial derivative, 450 principle of induction, 26 separable equation, 478, 530 separated, 435 separation, 438 sequence of functions, 334 sequence of partial products, 330 sequence of partial sums, 80, 295, 351 sequence, 15, 65 set, sided limit, 133 sigma, 21, 294 sign function, 154 simple continued fraction, 145 curve, 516 discontinuity, 159 region, 494 root, 213 simply connected, 506, 516 sinc function, 153, 227 sine integral, 273 singleton, 149 skew, 405 slope, 15 smooth, 397, 408 speed, 408, 410 sphere, 384,404 spherical Bessel function, 282 spherical coordinates, 431 spiral, 428 equiangular, 428 hyperbolic, 429 logarithmic, 428 logistic, 428 573 of Archimedes, 428 reciprocal, 429 squeeze theorem, 78, 120 stationary point, 196 Stieltjes, Jean, 240 Stirling, James, 277 Stirling's formula, 277, 288 Stokes, George, 519 Stokes' theorem, 519 streamline, 500 strictly increasing, 12 subadditive, 182 subinterval, 241 subrectangles, 481 subscript, 65 subsequence, 106 subsequential limit point, 107 subset, substitution, 260 sufficient, 32 sum, 295 sumrnable, 115, 116 summation by parts, 328 superadditive, 182 superposition principle, 475 supporting line, 239 supremum, 14,45 surface, 384, 404 surjection, 11 symmetric, 40 form, 402 interval, 19 w.r.t origin, 13 w.r.t point, 19 w.r.t vertical axis, 12 T tangent line, 184 plane, 470 vector, 408 tangential acceleration, 411 Tauber, Alfred, 364 Tauber's first theorem, 366 Tauberian hypothesis, 364 Tauberian theorems, 364 tautochrone, 399 Taylor, Brook, 213 Taylor coefficients, 214 Taylor polynomial, 214 Taylor series, 370 Taylor's formula, 214 Taylor's theorem, 214, 370, 468 telescoping, 80, 296 tends to infinity, 81, 121, 130, 133 terminal point, 385 Index 574 terms, 15, 65, 294 ternary set, 332 test for exactness, 528 theorem, 31 theory, 31 Thiele, Thorvald 147 Thomson, William, 519 three-dimensional system, 384 three-space, 384 tolerance, 67 topologist's sine curve, 448 topology, 433 Torricelli, Evangelista, 401 Torricelli's theorem, 401 total derivative, 458 trace, 405 traced, 396 transcendental, 46 transcendental function, 15, 135 transform, 289 transformation, 525 transitive, 40, 44 triangle inequality, 51, 390, 425 trichotomy, 44 trigonometric function, 12 twice differentiable, 413 two-dimensional, 6, 506 U unbounded, 13, 70 uncountable, 40 underbar, 244 uniform, 351 uniform continuity, 413 uniform limit, 341 uniform sum, 351 uniformly bounded, 339, 351 uniformly continuous, 170 uniformly differentiable, 212 union, uniqueness, 32 unit normal vector, 410 unit tangent vector, 408 unit vector, 386, 387 upper bound, 13,45 upper Darboux sum, 241 upper double integral, 482 upper integral, 244 upper limit, 107 upper sum, 241,481 V vanish, 14 variable, dependent, independent, variation, bounded, 236, 325 variation, total, 236 Varignon, Pierre, 429 vector, 385 acceleration, 448 addition, 386 difference, 386 differential operator, 459, 518 field, 500 function, 405 product, 392 triple product, 415 velocity, 408 vector-valued function, 397, 405 velocity, velocity vector field, 500 Venn diagram, Venn, John, vertex, vertical asymptote, 134 vertical line test, 8, 336 Viete, Franrois, 227 Volterra, Vito, 293 Volterra integro-differential equation, 293 w Wallis, John, 277 Wallis's formula, 277, 286 wave equation, 472 Weber, Heinrich, 279 Weber function, 279 Weber's equation, 283 Weierstrass intermediate value thm, 164 Weierstrass, Karl, 97 Weierstrass M-test, 354 weighted average, 64 weighted mean, 64 weights, 63, 64 well defined, 2, 10 well-ordering principle, 20 whole space, 436 Wiles, Andrew, 21 Witch of Agnesi, 340 work integral, 501 Wren, Christopher, 421 Wren's theorem, 421 Y Young, William, 371 z Zermelo, Ernst, 41 zero, 14, 212 of multiplicity, 213 vector, zeta function, 311, 331, 359 zipper sequence, 73